Queueing Systems With Fractional Number of Servers: Analysis and Practical Implementation of the Erlang-B Traffic Model

Abstract

In this paper, both the achievement and practical implementation of a fractional number of servers in queueing systems is proposed. The Erlang-B traffic model is considered to demonstrate how the fractional number of servers can be physically realized and incorporated into teletraffic analysis. While mathematical extensions of the Erlang-B formula to consider fractional number of servers have been explored in the literature, neither the practical realization nor the insensitivity property of such models has been extensively studied. In this paper, an approach is presented to effectively achieve fractional number of servers by considering, upon the arrival of service requests, the existence of $\lfloor s\rfloor +1$ servers at times, and only $\lfloor s\rfloor $ servers at other times. The strength of this approach lies in its easy integration into the Call Admission Control (CAC) strategy of real queueing systems. Utilizing a fractional number of servers enables precise adjustment of blocking probability, facilitating a fair and adequate performance comparison among different resource management strategies and evaluation scenarios. Additionally, the insensitivity property of the considered Erlang-B model with fractional number of servers is demonstrated. Finally, to illustrate the practical benefits and advantages of considering a fractional number of servers in the CAC and performance evaluation, numerical examples of the proposed concepts are provided and discussed, specifically within the context of a cellular system with VoIP traffic.

Keywords

• Call admission control
• Erlang-B formula
• fractional number of servers
• connection-level analysis
• blocking probability
• insensitivity property